Question

Find the derivative or f(x)= ax² + bx + c, where a,b,c are non-zero constant, by first principle.

Answer 1

Given,

f(x) = ax² + bx + c

∴ f'(x) = \(\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}\)

= \(\lim\limits_{h \to 0}\frac{[{a(x+h)^2+b(x+h)+c}]\, -\,[{ax^2+bx+c}]}{h}\)

= \(\lim\limits_{h \to 0}\frac{a(x^2+2xh+h^2)+bx+bh-ax^2-bx}{h}\)

= \(\lim\limits_{h \to 0}\frac{2axh+ah^2+bh}{h}\)

= 2ax + a∙0 + b

= 2ax + b